Lemma 9 (Lemma 7, [1]})
For all \(t\ge 1\) , the C-Tracking with weights \((\widetilde{{\omega }}(\widehat{{\mu }}_s))_{s\in \mathbb {N}^*}\)  ensures that \(N_a(t) \ge \sqrt{t+K^2} - 2K\)  and that
\(\max \limits _{1\le 1\le K} \bigg |N_a(t) - \sum \limits _{s=1}^t \widetilde{{\omega }}_a(\widehat{{\mu }}_s) \bigg | \le K(1+\sqrt{t})\) 

